Using Interpolation Theorem REF  (with \(d\nu _{1}=d\nu _{0}=W\left(x\right) dx\) ); Theorems 1 and 8 of [1]} and [2]}, if \(\varphi \in Q\) , \(W\in A_{p\left( \varphi \right) }\) , \(f\in L_{W}^{\varphi }\) , then the partial sum operator \(S_{n}:L_{W}^{\varphi }\rightarrow L_{W}^{\varphi }\)  (\(f\longmapsto S_{n}f\)  ) and conjugate
operator \(B:L_{W}^{\varphi }\rightarrow L_{W}^{\varphi }\)  (\(f\longmapsto \tilde{f}\)  ) are norm bounded in \(L_{W}^{\varphi }\) , namely,
\(\left\Vert S_{n}\left( f\right) \right\Vert _{\varphi ,W}\lesssim \left\Vert f\right\Vert _{\varphi ,W}\text{,\quad }\left\Vert Bf\right\Vert _{\varphi ,W}\lesssim \left\Vert f\right\Vert _{\varphi ,W}. \) 
